Herpes Virus Infection Probability Modeling

Project Overview

This project investigates the probability of herpes virus infection in cell cultures under controlled laboratory conditions. The goal is to model the likelihood of infection as a function of virus dilution.

In the experiments, cells were exposed to viral dilutions ranging from $10^{-1}$, $10^{-2}$, ..., $10^{-7}$ . After an incubation period, cytopathic effects (CPE) were assessed, which indicate successful infection. Each dilution was tested in 4 replicates per experiment to account for variability.

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Experimental Design

• Virus dilutions: $10^{-1}$, $10^{-2}$, ..., $10^{-7}$
• Replicates per dilution: 4 (currently)
• Outcome: Presence or absence of cytopathic effect (binary)

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Mathematical Model 1

We model the binary infection outcomes as following a Bernoulli distribution with parameter $p_i$:

$$y_i \sim \text{Bernoulli}(p_i)$$

where $p_i$ represents the probability of infection for observation $i$.

The infection probability $p_i$ is modeled using a logistic regression framework:

$$\text{logit}(p_i) = \alpha + \beta \cdot x_i$$

where:

• $x_i$ is the virus dilution (log₁₀ scale) for observation $i$
• $\alpha$ is the intercept (log-odds of infection when $x_i = 0$)
• $\beta$ is the fixed effect of virus dilution on log-odds of infection

In the Bayesian framework, we specify prior distributions for the parameters:

$$\alpha \sim \text{Normal}(\mu_\alpha, \sigma_\alpha)$$ $$\beta \sim \text{Normal}(\mu_\beta, \sigma_\beta)$$

Hierarchical Model for the Different Experiments

This section describes the hierarchical structure used to model the variation between different experiments in the project.

Each experiment $j$ has its own intercept $\alpha_j$ and slope $\beta_j$. These experiment-specific parameters are modeled hierarchically using a multivariate normal (MVN) distribution:

$$\begin{pmatrix} \alpha_j \\ \beta_j \end{pmatrix} \sim \text{MVN}\left( \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, \Sigma \right)$$

where:

$\alpha$ is the population-level (overall) intercept

$\beta$ is the population-level (overall) slope

$\Sigma$ is the $2 \times 2$ covariance matrix describing between-experiment variation

Covariance Matrix ($\Sigma$)

The covariance matrix $\Sigma$ is parameterized as:

$$\Sigma = \begin{pmatrix} \sigma_\alpha^2 & \rho , \sigma_\alpha \sigma_\beta \\ \rho , \sigma_\alpha \sigma_\beta & \sigma_\beta^2 \end{pmatrix}$$

where:

$\sigma_\alpha$ is the standard deviation of experiment-specific intercepts

$\sigma_\beta$ is the standard deviation of experiment-specific slopes

$\rho$ is the correlation between intercepts and slopes across experiments

Hyperparameter Priors

Population Mean Parameters:

$$\alpha \sim \text{Normal}(\mu_\alpha, \sigma_\alpha)$$

$$\beta \sim \text{Normal}(\mu_\beta, \sigma_\beta)$$

Population Standard Deviation Parameters (Scale):

$$\sigma_\alpha \sim \text{Half-Normal}(0, \tau_\alpha)$$

$$\sigma_\beta \sim \text{Half-Normal}(0, \tau_\beta)$$

Correlation Parameter:

$$\rho \sim \text{LKJCorr}(2)$$